J Intell Manuf (2011) 22:815–822 DOI 10.1007/s10845-009-0370-y Ant colony optimization algorithm for a Bi-criteria 2-stage hybrid flowshop scheduling problem Elyn L. Solano-Charris · Jairo R. Montoya-Torres · Carlos D. Paternina-Arboleda Received: 24 January 2009 / Accepted: 14 December 2009 / Published online: 31 December 2009 © Springer Science+Business Media, LLC 2009 Abstract We consider the problem of scheduling jobs in a hybrid flowshop with two stages. Our objective is to min- imize both the makespan and the total completion time of jobs. This problem has been little studied in the literature. To solve the problem, we propose an ant colony optimization procedure. Computational experiments are conducted using random-generated instances from the literature. In compari- son against other well-known heuristics from the literature, experimental results show that our algorithm outperforms such heuristics. Keywords Scheduling · Hybrid flowshop · Ant colony optimization · Multi-criteria Introduction The theory of scheduling is generally defined as a decision- making process with the aim of executing a set of jobs on limited resources by optimizing one or more performance criteria. Flowshop scheduling is one of the most studied com- binatorial optimization problems in scheduling theory. It con- sists on a set of n jobs that have to be processed on m machines in series. All jobs have the same processing routing: first E. L. Solano-Charris (B ) · J. R. Montoya-Torres (B ) Escuela Internacional de Ciencias Económicas y Administrativas, Universidad de La Sabana, Autopista norte de Bogotá D.C., Chía (Cundinamarca), Colombia e-mail: elynsolano_@hotmail.com J. R. Montoya-Torres e-mail: jairo.montoya@unisabana.edu.co C. D. Paternina-Arboleda Departamento de Ingeniería Industrial, Universidad del Norte, km 5 vía Puerto Colombia, Barranquilla, Colombia e-mail: cpaterni@uninorte.edu.co machine, second machine, and so on. This paper focuses on a general configuration called the flexible or hybrid flowshop scheduling (HFS) problem. In the HFS configuration, a set of n jobs have to be processed on k stages in series, each stage containing a set of parallel identical machines. The hybrid flowshop scheduling configuration is very often found in sev- eral real-life manufacturing environments, such as electronic manufacturing environment such as IC (Integrated Circuit) packaging and make-to-stock wafer manufacturing. Formally, we consider the hybrid flowshop scheduling (HFS) problem defined as follows. There is a set of n jobs that has to be processed on k = 2 stages in series. At stage l,l = 1, 2, there are M (l) identical machines in parallel. There is unlimited intermediate storage between two successive stages. Job J i ,i = 1,...,n has to be processed only once at each stage on any one machine. The processing times of job J i at each stage are denoted as p (1) i and p (2) i , respec- tively. The goal is to find a schedule without pre-emption that minimizes some performance criteria. Figure 1 shows an example of job routing in a 2-stage HFS and variable number of machines at each stage, in which arrows represent the processing routing a given job. Since the two-stage hybrid flowshop scheduling problem with makespan minimization is NP-hard (Gupta 1988). In the literature, most of the work on hybrid flowshop problems focuses on mono-objective two-stage problems. The most studied objective function is the makespan (Reza Hejazi and Saghafian 2005). Several exact methods as well as approximation algorithms and dedicated heuristics have been proposed, e.g. Brah and Hunsucker (1991), Chen (1995), Haouari and M’Hallah (1997), Dessouky et al. (1998), Moursli and Pochet (2000). State-of-the-art surveys are pro- posed in Chen (1994), Linn and Zhang (1999). While inter- esting results have been obtained for the two-stage hybrid flowshop scheduling problems, there has been less work on 123